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This conjecture is true. Of course, the hardest part of this problem is the presence of the $\min$. So, the crucial point in the proof is the following upper bound:
Lemma 1. \begin{equation*} \frac1{\min(m+q-1,x)}\le\frac Am+\frac Bx, \end{equation*} where \begin{equation*} A:=\frac{(x-q+1)^2}{(x+q-1)^2},\quad B:=1-A=\frac{4 (q-1) x}{(x+q-1)^2}, \tag{3} \end{equation*} \begin{equation*} 1<q\le x,\quad 1\le m\le x. \tag{4} \end{equation*}
In what follows, (3) and (4) will always be assumed. (Recall that the case $q=1$ was already verified by the OP, and it will also follow by continuity.) The proof of Lemma 1 is quite elementary, but a bit tedious, and omitted here.
By Lemma 1, the left-hand side (lhs) of the inequality in question is upper bounded by \begin{equation*} \sum_{m=1}^{x} m\,\Big(\frac Am+\frac Bx\Big) \binom{x}{m} \left( \frac{x-q}{x+q}\right)^m =A (r-1)+\frac{B r (x-q)}{2 x}, \end{equation*} where \begin{equation*} r:=\left(\frac{2x}{x+q}\right)^x. \end{equation*} So, the inequality in question reduces to \begin{equation*} A (r-1)+\frac{B r (x-q)}{2 x}\le \frac{x-q+2}{x+q}\,(r-1), \end{equation*} which can be rewriten as \begin{equation} \de(q):=\de(q,x):=\ln r-\ln\frac{(x+1)^2-1-(q-1)^2}{2 x + 2 q-1}\ge0. \tag{5} \end{equation} It is straightforward but tedious to see that \begin{multline*} [(x+1)^2-1-(q-1)^2]\de'(q) \\ =2 q^3 (x+1)+q^2 \left(2 x^2+x-2\right)-2 q x \left(x^2+x-1\right)-x \left(2 x^3+x^2-4 x+1\right)<0 \end{multline*} for \begin{equation} 1\le q\le x-2/5. \tag{6} \end{equation} So, $\de(q)=\de(q,x)$ is decreasing in $q\in[1,x-2/5]$. Moreover, \begin{multline*} (5 x-1) (20 x-9) ( 120 x-49) \frac d{dx}\de(x-2/5,x) \\ =12000 x^3 \ln (5)-100 x^2 (24+127 \ln (5)) \\ +\left(12000 x^3-12700 x^2+4265 x-441\right) \ln \left(\frac{x}{5 x-1}\right)\\ +5 x (512+853 \ln (5))-541-441 \ln (5). \end{multline*} It is straightforward but tedious to see that the latter expression is positive for all $x\ge1$. (One way to deal with such an expression is to notice that the derivative of a high enough order of an expression of the form $P(x)\ln R(x)$ is a polynomial if $P(x)$ is a polynomial and $R(x)$ is the ratio of polynomials.) So, $\de(x-2/5,x)$ is increasing in $x\ge1$. Also, $\de(x-2/5,x)|_{x=2}=0.00187\ldots>0$. So, $\de(x-2/5,x)>0$ for $x\ge2$. Recalling that $\de(q)=\de(q,x)$ is decreasing in $q\in[1,x-2/5]$, we see that $\de(q,x)>0$ for $x\ge2$ and $q\in[1,x-2/5]$.
Thus, we get the inequality in question for $x\ge2$ and $q\in[1,x-2/5]$. The case $x=1$ is trivial.
So, it remains to consider the case when $x\ge2$ and $q\in(x-2/5,x]$. This case is much easier than the one considered, because in this case the $\min$ does not cause trouble, because then $\min(m+q-1,x)=x$, a constant, for $m=2,\dots,x$. So, in this case the difference between the left- and right-hand sides of the inequality in question times $2 q x (x+q)$ equals $2 x (x^2 - q (x - 2)) - q ((x - q)^2 + 4 x)r$, with $r$ as before. Hence, the inequality in question can be rewritten here as \begin{equation} \de(q):=\de(q,x):=\ln r-\ln\frac{2 x (x^2 - q (x - 2))} {q r ((x - q)^2 + 4 x)}\ge0; \tag{7} \end{equation} here we use the same notation, $\de(q)=\de(q,x)$, for an expression (somewhat similar to but) different from the one in (5). Here, we have \begin{multline*} q (x + q) (x^2 - q (x - 2)) ((x - q)^2 + 4 x)\de'(q) \\ =q^3 (11 - 6 x) x^2 + x^4 (4 + x) + q x^3 (4 - 11 x - 2 x^2) \\ + 2 q^4 (2 - 3 x + x^2) + q^2 x^2 (-20 + 5 x + 6 x^2)<0 \end{multline*} for all $q\in[1,x]$, so that $\de(q,x)$ is decreasing in $q\in[1,x]$. Also, $\de(x,x)=0$ and hence $\de(q,x)>0$ for all $q\in[1,x]$.
Thus, the inequality in question is completely proved.